Cleanness of skew generalized power series rings
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A skew generalized power series ring $R[[S, \omega]]$ consists of all functions from a strictly ordered monoid $S$ to a ring $R$ whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action $\omega$ of the monoid $S$ on the ring $R$. Special cases of the skew generalized power series ring construction are skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Mal'cev-Neumann series rings, the ``untwisted" versions of all of these, and generalized power series rings. In this paper we obtain some necessary conditions on $R$, $S$ and $\omega$ such that the skew generalized power series ring $R[[S,\omega ]]$ is (uniquely) clean. As particular cases of our general results we obtain new theorems on skew Mal'cev-Neumann series rings, skew Laurent series rings, and generalized power series rings.
Abstract
A skew generalized power series ring $R[[S, \omega]]$ consists of all functions from a strictly ordered monoid $S$ to a ring $R$ whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action $\omega$ of the monoid $S$ on the ring $R$. Special cases of the skew generalized power series ring construction are skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Mal'cev-Neumann series rings, the ``untwisted" versions of all of these, and generalized power series rings. In this paper we obtain some necessary conditions on $R$, $S$ and $\omega$ such that the skew generalized power series ring $R[[S,\omega ]]$ is (uniquely) clean. As particular cases of our general results we obtain new theorems on skew Mal'cev-Neumann series rings, skew Laurent series rings, and generalized power series rings.
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